Degree 2 to Degree 4!!!

Algebra Level 3

Given that: x 2 − 5 x + 1 = 0 x^2-5x+1=0

Find the value of x 4 + 1 x 4 x^4+\frac{1}{x^4}


The answer is 527.

Mj Santos by MJ Santos , 22, Philippines

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1 solution

Mj Santos
Feb 1, 2015

Note that: x 2 − 5 x + 1 = 0 ⇒ x 2 + 1 = 5 x x^2-5x+1=0 \Rightarrow x^2+1=5x

Dividing both sides by x gives:

x + 1 x = 5 x+\frac{1}{x}=5

Square both sides:

x 2 + 1 x 2 + 2 = 5 2 ⇒ x 2 + 1 x 2 = 23 x^2+\frac{1}{x^2}+2=5^2 \Rightarrow x^2+\frac{1}{x^2}=23

Square again both sides to have a degree 4:

x 4 + 1 x 4 + 2 = 23 2 ⇒ x 4 + 1 x 4 = 529 − 2 = 527 x^4+\frac{1}{x^4}+2={23}^2 \Rightarrow x^4+\frac{1}{x^4} = 529-2 =\boxed{527}

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